Algebra - Washington State University

Algebra

1. (a + b)2 = a2 + 2ab + b2 2. (a - b)2 = a2 - 2ab + b2 3. a2 - b2 = (a - b)(a + b)

4. ax2 + bx + c =

-b + b2 - 4ac a x-

2a

if b2 - 4ac 0

-b - b2 - 4ac

x-

,

2a

Trigonometry

Definitions

sin x 1. tan x =

cos x cos x 2. cot x = sin x

Identities

1. cos2 x + sin2 x = 1

2. sin 2x = 2 sin x cos x

3. cos 2x = = cos2 x - sin2 x = 2 cos2 x - 1 = 1 - 2 sin2 x

1 3. sec x =

cos x 1

4. csc x = sin x

4. tan2 x + 1 = sec2 x 5. cot2 x + 1 = csc2 x

2 tan x 6. tan 2x = 1 - tan2 x

cot2 x - 1 7. cot 2x =

2 cot x

Hyperbolic functions

Definitions

ex - e-x

1. sinh x = 2

1 3. csch x =

sinh x sinh x 5. tanh x = cosh x

ex + e-x

2. cosh x = 2

1 4. sech x =

cosh x cosh x 6. coth x = sinh x

Identities 1. cosh2 x - sinh2 x = 1

4. sinh 2x = 2 sinh x cosh x

2. 1 - tanh2 x = sech2 x 3. coth2 x - 1 = csch2 x

5. cosh 2x = = cosh2 x + sinh2 x = 2 cosh2 x - 1 = 2 sinh2 x + 1

Derivatives

Derivatives

1. (sin x) = cos x 2. (cos x) = - sin x 3. (tan x) = sec2 x 4. (cot x) = - csc2 x

1 7. (arcsin x) =

1 - x2 -1 8. (arccos x) = 1 - x2

5. (sec x) = tan x sec x 1

6. (csc x) = - cot x csc x 9. (arctan x) = 1 + x2

Integrals

1. (sinh x) = cosh x

2. (cosh x) = sinh x 3. (tanh x) = sech2 x 4. (coth x) = - csch2 x

5. (sech x) = = - tanh x sech x

6. (csch x) = = - coth x csch x

7. (arcsinh x) = 1

= x2 + 1

8. (arccosh x) = 1

= x2 - 1

9. (arctanh x) = 1

= x2 - 1

1. cos x dx = sin x

6. tan x dx =

2. sin x dx = - cos x

= - ln | cos x|

3. sec2 x dx = tan x

7. sec x dx = = ln | tan x + sec x|

4. csc2 x dx = - cot x 8. csc x dx =

5. cot x dx = ln | sin x|

= - ln | cot x + csc x|

Symmetry & Periodicity 1. sin(-x) = - sin(x) 2. sin(x + /2) = cos(x) 3. sin(x - /2) = - cos(x) 4. sin(x ? ) = - sin(x)

5. cos(-x) = cos(x) 6. cos(x + /2) = - sin(x) 7. cos(x - /2) = sin(x) 8. cos(x ? ) = - cos(x)

9. tan(-x) = - tan(x) 10. tan(x + /2) = - cot(x) 11. tan(x - /2) = - cot(x) 12. tan(x ? ) = tan(x)

x

- -/2 0 /6 /4 /3 /2 2/3 5/6 3/2 2

sin(x) 0

-1 0

1 2

1 2

3 2

1

3 2

1 2

0 -1 0

cos(x) -1

0

1

3 2

1 2

1 2

0

-

1 2

-

3 2

-1

0

1

tan(x) 0

?

0 1

1

3

?

-3

- 1

0

?

?

3

3

cot(x) ?

0

?3

1

1

1

- 1

-3

?

0

1

3

3

Powers, Exponents & Logarithms

1. xa+b = xa ? xb

2.

xa-b =

xa xb

3.

x-a =

1 xa

4. xa?b = (xa)b

5. xa/b = b xa

6. ex : (-, ) (0, ) 7. eln(x) = x 8. ln(ex) = x

Derivatives & Integrals 1. (xn) = nxn-1

3. (eax) = aeax

2.

xn

dx =

xn+1 ,

n = -1

n+1

4. eax dx = eax

a

9. ln(x) : (0, ) (-, )

10. ln(a ? b) = ln(a) + ln(b) a

11. ln = ln(a) - ln(b) b

12. ln(ab) = b ln(a)

1 5. (ln(x)) =

x

dx

6.

= ln(x)

x

7. ln(x) dx = x ln(x) - x

Other Integrals

1

1

x

1. x2 + a2 dx = a arctan |a|

2. 1 dx = ln |x + x2 ? a2| x2 ? a2

Inverse trigonometric functions

arcsin x = arccos 1 - x2

sin(y) cos(y) tan(y) cot(y)

y = arcsin(x) x

1 - x2 x

1 - x2

1 - x2 x

y = arccos(x)

1 - x2

x

1 - x2 x x

1 - x2

y = arctan(x) x

x2 - 1 1

1 + x2

x

1 x

y = arccot(x) 1

1 + x2 x

x2 - 1 1 x

x

Inverse hyperbolic functions

sinh(y) cosh(y) tanh(y) coth(y)

y = arcsinh(x) x

x2 + 1 x

x2 + 1

x2 + 1 x

y = arccosh(x) x2 - 1

x

x2 - 1 x x

x2 - 1

y = arctanh(x) x

1 - x2 1

1 - x2

x

1 x

y = arccoth(x) sign x

x2 - 1 |x|

x2 - 1 1 x

x

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